Resource Analysis of a Fault-Tolerant Quantum Computer
Yoshihisa Yamamoto, Cody Jones, Rodney Van Meter
最先端研究開発支援プログラム「量子情報処理プロジェクト」全体会議2011
(京都 2011年12月13-16日)
Problem Statement
Slow down by ~108
Optical control
of quantum dot spins
Logical Layer
QEC Layer
Virtual Layer
Physical Layer
3
Physical Qubits - Cavity QED Systems with Single-Electron-Doped Quantum Dots -
A simple planar microcavity with
2D lattice of site-controlled QDs C. Schneider, M. Strauss, T. Sunner, A. Huggenberger,
D. Wiener, S. Reitzenstein, M. Kamp, S. Höfling and A.
Forchel APL 92, 183101 (2008)
N. Cody Jones, Rodney Van Meter, Austin G. Fowler, Peter L. McMahon,
Jungsang Kim, Thaddeus D. Ladd, and Yoshihisa Yamamoto
arXiv:1010.5022v2 [quant-ph] 5 Dec 2011
Layered architecture for quantum computing
Primary Components of the Physical Layer
4
(Stimulated Raman scattering)
Proposal: PRL 99, 040501(2007), PRB 84, 235307(2011)
Electron spin: Nature 456, 218(2008), Nature Physics 4, 780(2008), Nature Photonics 4, 367(2010)
Hole spin: Nature physics 7, 872(2011)
(Optical Kerr rotation)
(geometric phase gate)
5
Parameters for Layer 1: Quantum Operation
4 ps
Virtual Layer
Cause destructive interference of systematic errors
Dynamical decoupling
sequence to correct
dephasing errors • T2
* is very fast, so embed DD
at lowest level
BB1 compensation sequence to correct gate errors, such as
laser intensity fluctuations • Wimperis, J. Magn. Reson. Ser. B 109, 221 (1994)
7
Virtual Layer
A special dynamical decoupling sequence for QuDOS, known as
8H since it requires eight Hadamard pulses. TL is the Larmor
period determined by the external magnetic eld. (a) Timing
specication for the 8H sequence, where is an arbitrary time.
Each of the pulse pairs enacts a -rotation around the X-axis of
the virtual qubit Bloch sphere. For 8H to work eciently, T2. (b)
Four 8H sequences in a row interleaved with arbitrary gates
formed from three Hadamard pulses (orange). The overall
sequence forms a virtual gate by way of a BB1 compensation
sequence.
Simulation of the decoupling eectiveness of the 8H
sequence compared to CP and UDD (each using 4 X
gates) in the presence of dephasing noise and control
errors. Here, “pulse error” is a systematic, relative
deviation in the energy of every pulse. In all cases,
two Hadamard pulses are combined to produce an
approximate X gate. The vertical axis is indelity after
evolution of the sequence in Fig. (a) with = 1 ns;
here indelity is 1 - F = 1 - II , where II is the
identity-to-identity matrix element in process
tomography for the decoupling gate
sequence with random noise. Since we aim to
execute virtual gates with 1-F < 10-3, laser pulse
errors must be less than 1% in order for the virtual
qubit memory error rate to be adequately low.
8
Optical Spin Echo Experiment with a Single Spin in Planar Microcavity D. Press et al., Nature Photonics 4, 367 (2010)
Shorter T2 observed at B<4T
( paramagnetic impurity?)
Fowler, et al. Phys. Rev. A 80, 052312 (2009)
Fowler, et al. arXiv/1110.5133 (2011)
Quantum Error Correction Layer
Surface code: estimate distance needed
Extract Syndrome
Syndrome Matching
εthresh 9×10-3
εV 10-3
C 0.03
εL 10-15
d 29
Estimating Surface Code Distance
10
(detect errors)
(store information)
Topological Surface Code
A lattice refresh cycle of the surface code can be performed in parallel
across the entire 2D array of qubits.
Two advantages:
• High threshold for gate errors 1 %
• Requires only nearest-neighbor interaction
Separation in Time Scales
Operation times increase by orders of magnitude from Physical to Logical layer
12
Primary Control Cycle of the Surface Code Quantum Computer - Pipelining -
N. Cody Jones et al., arXiv:1010.5022 [quant-ph]
13
# of logical qubits Q ~ 6N
Success probability of quantum
algorithm
~99.99%
L≤10-4/KQ
code distance 53
total number of physical
qubits
2 x 108 (minimum)
~ 109 (sufficient)
Shor’s factoring algorithm for 2048-bit integer
Resource and Operation Time of Surface Code Architecture
Total computational time ~10 days
Circuit Depth Shor’s Algorithm
Assumptions
Optical quantum dots
Surface code QEC
Shor implementation given in
[Van Meter, et al. IJQI 8, 295 (2010)]
Fixed size: 105 logical qubits
• Algorithm stalls when distillation is not fast enough
• Require ~90% of QC devoted to distillation
15 誤り訂正コード
量子ビット
デコヒーレンス
量子誤り訂正コードで量子ビットを守る - 過保護に育てられたひ弱な子供のように -